Syrian refugee children near the Syria-Turkey border during April 2012. Photograph: Ma Yan/Xinhua Press/Corbis

Syria's humanitarian crisis is spinning out of control as refugee numbers spike and funding to help ease the suffering has yet to materialise, Oxfam has said.

Oxfam's Syria crisis response manager, Francis Lacasse, warned that the "worst-case scenario" for the UN, of more than a million refugees from Syria being based in other countries by this June, could be realised within weeks.

Each day about 5,000 refugees are fleeing Syria to neighbouring countries, such as Jordan, Lebanon and Turkey – a 36% increase compared with the December 2012 figures, Lacasse said in a news release.

He warned that the surge was placing a massive burden on the countries, with the potential to undermine stability in the region.

"The humanitarian crisis is worsening day by day, leaving agencies struggling to provide help that's desperately needed," Lacasse said.

He also said that only 20% of the $1.5m aid pledged by the US, other western nations and Gulf Arab countries, at a donor conference last month in Kuwait, had been received.

Lacasse said the money was "urgently needed now" to allow agencies to continue providing basic services such as food, water and shelter to the ever-growing refugee populations.

Jordanian officials said they had seen a new surge in Syrians fleeing across the border into their country in recent days, as fighting intensified in southern Syria.

This month, Jordan recorded its highest-ever influx of refugees, with more than 50,000 new arrivals.

Anmar Hmoud, a government spokesman for Syrian refugee affairs, said Jordan was now hosting 418,529 Syrian refugees. The UN said that that number could rise to half a million by the end of March.